A048733 -id:A048733 - cp's OEIS frontend

A048715 Binary expansion matches (100(0))(0|1|10)?; or, Zeckendorf-like expansion of n using recurrence f(n) = f(n-1) + f(n-3).

0, 1, 2, 4, 8, 9, 16, 17, 18, 32, 33, 34, 36, 64, 65, 66, 68, 72, 73, 128, 129, 130, 132, 136, 137, 144, 145, 146, 256, 257, 258, 260, 264, 265, 272, 273, 274, 288, 289, 290, 292, 512, 513, 514, 516, 520, 521, 528, 529, 530, 544, 545, 546, 548, 576, 577, 578, 580

Offset: 0

Antti Karttunen, Mar 30 1999

No more than one 1-bit in each bit triple.
All terms satisfy A048727(n) = 7*n.
Constructed from A000930 in the same way as A003714 is constructed from A000045.
It appears that n is in the sequence if and only if C(7n,n) is odd (cf. A003714). - Benoit Cloitre, Mar 09 2003
The conjecture by Benoit is correct. This is easily proved using the well-known result that the multiplicity with which a prime p divides C(n+m,n) is the number of carries when adding n+m in base p. - Franklin T. Adams-Watters, Oct 06 2009
Appears to be the set of numbers x such that (x AND 5*x) = x and (x OR 3*x)/x = 3. - Gary Detlefs, Jun 08 2024

G. C. Greubel, Table of n, a(n) for n = 0..1275
Sebastian Karlsson, Walnut code that verifies the conjectures of Paul D. Hanna
Walnut can be downloaded from https://cs.uwaterloo.ca/~shallit/walnut.html.
Index entries for sequences defined by congruent products between domains N and GF(2)[X]
Index entries for sequences defined by congruent products under XOR
Index entries for 2-automatic sequences.

Subsequence of A048716.

Cf. A003726, A004742, A004743, A004744, A048717, A048718, A048719, A048730, A048733, A115422, A115423, A115424.

Reap[Do[If[OddQ[Binomial[7n, n]], Sow[n]], {n, 0, 400}]][[2, 1]]
(* Second program: *)
filterQ[n_] := With[{bb = IntegerDigits[n, 2]}, !MatchQ[bb, {_, 1, 0, 1, _}|{_, 1, 1, _}]];
Select[Range[0, 580], filterQ] (* Jean-François Alcover, Dec 31 2020 *)

is(n)=!bitand(n, 6*n) \\ Charles R Greathouse IV, Oct 03 2016

for my $k (0..580) { print "$k, " if sprintf("%b", $k) =~ m{^(100(0)*)*(0|1|10)?$}; } # Georg Fischer, Jun 26 2021

import re
def ok(n): return re.fullmatch('(100(0)*)*(0|1|10)?', bin(n)[2:]) != None
print(list(filter(ok, range(581)))) # Michael S. Branicky, Jun 26 2021

a(0) = 0, a(n) = (2^(invfoo(n)-1))+a(n-foo(invfoo(n))), where foo(n) is foo(n-1) + foo(n-3) (A000930) and invfoo is its "integral" (floored down) inverse.
a(n) XOR 6*a(n) = 7*a(n); 3*a(n) XOR 4*a(n) = 7*a(n); 3*a(n) XOR 5*a(n) = 6*a(n); (conjectures). - Paul D. Hanna, Jan 22 2006
The conjectures can be verified using the Walnut theorem-prover (see links). - Sebastian Karlsson, Dec 31 2022

Definition corrected by Georg Fischer, Jun 26 2021

A048717 Binary expansion matches ((0)00(1)11)(0).

Original entry on oeis.org

0, 3, 6, 7, 12, 14, 15, 24, 28, 30, 31, 48, 51, 56, 60, 62, 63, 96, 99, 102, 103, 112, 115, 120, 124, 126, 127, 192, 195, 198, 199, 204, 206, 207, 224, 227, 230, 231, 240, 243, 248, 252, 254, 255, 384, 387, 390, 391

Offset: 0

Antti Karttunen, Mar 30 1999

In binary expansion, 1-bits occur only in groups of two or more, separated from other such groups by at least two 0-bits.

Integers that satisfy A048727(n) = 3*n.

Index entries for sequences defined by congruent products between domains N and GF(2)[X]

Row 3 of A115872. Superset of A048719. Cf. A048733.

filterQ[n_] := !MatchQ[IntegerDigits[n, 2], {1}|{1, 0, _}|{_, 0, 1}|{_, 1, 0, 1, _}|{_, 0, 1, 0, _}];
Select[Range[0, 400], filterQ] (* Jean-François Alcover, Dec 31 2020 *)

A048730 Differences between A008589 (multiples of 7) and A048727, a(n) = ((n*7)-Xmult(n,7)).

Original entry on oeis.org

0, 0, 0, 12, 0, 8, 24, 28, 0, 0, 16, 28, 48, 56, 56, 60, 0, 0, 0, 12, 32, 40, 56, 60, 96, 96, 112, 124, 112, 120, 120, 124, 0, 0, 0, 12, 0, 8, 24, 28, 64, 64, 80, 92, 112, 120, 120, 124, 192, 192, 192, 204, 224, 232, 248

Offset: 0

Antti Karttunen, Apr 26 1999

For n = binary n[k],n[k-1],...,n[0], bits a(n) = binary b[k+1],b[k],...,b[0] are b[i] = 1 when n[i-1] + n[i-2] + n[i-3] >= 2, so the majority bit 0 or 1 among the 3 bits of n below position i (with 0 bits below the radix point of n as necessary). This is since 7*n = 4*n + 2*n + n is n[i-1] + n[i-2] + n[i-3] at position i-1, and 4*n XOR 2*n XOR n is the same but no carry, so b[i] is the carry only. - Kevin Ryde, Mar 26 2021

Paolo Xausa, Table of n, a(n) for n = 0..8191

Positions of zeros are given by A048715. Cf. A048733, A342697.

Diagonal 7 of A061858.

A048730[n_] := 7*n - BitXor[n, 2*n, 4*n];
Array[A048730,100, 0] (* Paolo Xausa, Aug 06 2025 *)

a(n)=7*n - bitxor(n, bitxor(2*n, 4*n)) \\ Charles R Greathouse IV, Oct 03 2016

def A048730(n): return 7*n-(n^n<<1^n<<2) # Chai Wah Wu, Jun 29 2022

A342697 For any number n with binary expansion Sum_{k >= 0} b(k) * 2^k, the binary expansion of a(n) is Sum_{k >= 0} floor((b(k) + b(k+1) + b(k+2))/2) * 2^k.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 3, 3, 0, 0, 2, 3, 6, 7, 7, 7, 0, 0, 0, 1, 4, 5, 7, 7, 12, 12, 14, 15, 14, 15, 15, 15, 0, 0, 0, 1, 0, 1, 3, 3, 8, 8, 10, 11, 14, 15, 15, 15, 24, 24, 24, 25, 28, 29, 31, 31, 28, 28, 30, 31, 30, 31, 31, 31, 0, 0, 0, 1, 0, 1, 3, 3, 0, 0, 2, 3, 6

Offset: 0

Rémy Sigrist, Mar 18 2021

The value of the k-th bit in a(n) corresponds to the most frequent value in the bit triple starting at the k-th bit in n.

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     0       1          0
   2     0      10          0
   3     1      11          1
   4     0     100          0
   5     1     101          1
   6     3     110         11
   7     3     111         11
   8     0    1000          0
   9     0    1001          0
  10     2    1010         10
  11     3    1011         11
  12     6    1100        110
  13     7    1101        111
  14     7    1110        111
  15     7    1111        111

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Michael De Vlieger, Log log scatterplot of a(n), n = 0..2^20.
Index entries for sequences related to binary expansion of n

Cf. A048715, A048730, A048733, A342698, A342700.

A342697[n_] := Quotient[7*n - BitXor[n, 2*n, 4*n], 8];
Array[A342697, 100, 0] (* Paolo Xausa, Aug 06 2025 *)

a(n) = sum(k=0, #binary(n), ((bittest(n, k)+bittest(n, k+1)+bittest(n, k+2))>=2) * 2^k)

a(n) = 0 iff n belongs to A048715.

a(n) = floor(A048730(n)/8) = floor(A048733(n)/2). - Kevin Ryde, Mar 26 2021

cp's OEIS Frontend

A048715 Binary expansion matches (100(0)*)*(0|1|10)?; or, Zeckendorf-like expansion of n using recurrence f(n) = f(n-1) + f(n-3).

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A048717 Binary expansion matches ((0)*00(1*)11)*(0*).

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A048730 Differences between A008589 (multiples of 7) and A048727, a(n) = ((n*7)-Xmult(n,7)).

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A342697 For any number n with binary expansion Sum_{k >= 0} b(k) * 2^k, the binary expansion of a(n) is Sum_{k >= 0} floor((b(k) + b(k+1) + b(k+2))/2) * 2^k.

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A048715 Binary expansion matches (100(0))(0|1|10)?; or, Zeckendorf-like expansion of n using recurrence f(n) = f(n-1) + f(n-3).

A048717 Binary expansion matches ((0)00(1)11)(0).